The Cauchy Problem for Higher Order Equations

1971 ◽  
pp. 48-86
Author(s):  
Fritz John
Keyword(s):  
Cauchy Problem ◽  
Higher Order ◽  
The Cauchy Problem

scholarly journals The Cauchy Problem for Higher-Order KP Equations

1999 ◽  
Vol 153 (1) ◽  
pp. 196-222 ◽  
Author(s):  
J.C. Saut ◽  
N. Tzvetkov
Keyword(s):  
Cauchy Problem ◽  
Higher Order ◽  
The Cauchy Problem ◽  
Kp Equations

scholarly journals The Cauchy problem for higher-order modified Camassa–Holm equations on the circle

2019 ◽  
Vol 187 ◽  
pp. 397-433 ◽  
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Xiaoping Zhai ◽  
Yimin Zhang
Keyword(s):  
Cauchy Problem ◽  
Higher Order ◽  
The Cauchy Problem

scholarly journals On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohammad Kafini
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Blow Up ◽  
Initial Energy ◽  
Higher Order ◽  
Viscoelastic Wave Equation ◽  
Nonlinear Viscoelastic ◽  
Viscoelastic Wave ◽  
The Cauchy Problem ◽  
Whole Space

<p style='text-indent:20px;'>In this paper we consider the Cauchy problem for a higher-order viscoelastic wave equation with finite memory and nonlinear logarithmic source term. Under certain conditions on the initial data with negative initial energy and with certain class of relaxation functions, we prove a finite-time blow-up result in the whole space. Moreover, the blow-up time is estimated explicitly. The upper bound and the lower bound for the blow up time are estimated.</p>

HIGHER-ORDER QUASILINEAR PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA

2006 ◽  
Vol 08 (03) ◽  
pp. 331-354 ◽  
Author(s):  
V. A. GALAKTIONOV ◽  
A. E. SHISHKOV
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Quasilinear Parabolic Equation ◽  
Higher Order ◽  
Quasilinear Parabolic Equations ◽  
Semilinear Heat Equation ◽  
Basic Model ◽  
The Cauchy Problem ◽  
Singular Initial Data ◽  
Quasilinear Parabolic

As a basic model, we study the 2mth-order quasilinear parabolic equation of diffusion-absorption type [Formula: see text] where Δm,p is the 2mth-order p-Laplacian [Formula: see text]. We consider the Cauchy problem in RN × R+ with arbitrary singular initial data u0 ≠ 0 such that u0(x) = 0 for any x ≠ 0. We prove that, in the most delicate case p = q and [Formula: see text], this Cauchy problem admits the unique trivial solution u(·, t) = 0 for t > 0. For λ < λ0, such nontrivial very singular solutions are known to exist for some semilinear higher-order models. This extends the well-known result by Brezis and Friedman established in 1983 for the semilinear heat equation with p = q = m = 1.

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